Stokes theorem for smooth singular chains

For one smooth simplex $\sigma$, take a neighbourhood extension $\bar{\sigma }$ as required in [F2]. The smooth form $\eta ={\bar{\sigma }}^{\ast }\omega$ is defined on that whole neighbourhood in the affine span, so [F1] applies. By [F4] its derivative is ${\bar{\sigma }}^{\ast }d\omega$. By [F5], its pullback to face $i$ is $(\bar{\sigma }\circ {\delta }_i{)}^{\ast }\omega$, which extends the face simplex smoothly. Definition [F3] consequently gives ${\int }_{\sigma }d\omega ={\sum }_{i=0}^k(-1{)}^i{\int }_{\sigma \circ {\delta }_i}\omega \mathrm{.}$ The values do not depend on the selected extension.

A chain is a finitely supported coefficient function on the supplied simplex set. Define ${\int }_c\omega ={\sum }_{\sigma \in \mathrm{supp}c}c(\sigma ){\int }_{\sigma }\omega$. This is independent of a written expression for $c$: combining repetitions adds their coefficients, and inserting a zero coefficient changes nothing. Finite distributivity gives additivity and real homogeneity in $c$. No basis selection or simultaneous choice of extensions is involved, because [F3] already assigns a unique value to each simplex.

Multiply step 1.1 by $c(\sigma )$ and sum over its finite support. The boundary from [F2] is the same finite double sum of face simplices with coefficients $c(\sigma )(-1{)}^i$. When identical faces occur, step 1.2 combines their coefficients in exactly the same way in its integral. Therefore ${\int }_{c}d\omega ={\sum }_{\sigma ,i}c(\sigma )(-1{)}^i{\int }_{\sigma {\delta }_i}\omega ={\int }_{\mathrm{\partial }c}\omega \mathrm{.}$

For $k=1$, [F1] uses the terminal-minus-initial endpoint evaluations, so the formula includes every smooth path, constant or otherwise. Degenerate higher simplices remain generators, and step 1.1 applies to their smooth extensions without a rank assumption. If $c=0$ or $M=\mathrm{\varnothing }$, the relevant sums are empty and both integrals vanish. A zero form has zero pullback and zero integral. The assertion involves only positive chain degrees and finite sums, so no negative degree or choice assumption is hidden.